There is absolutely, obviously, still private property in the Federation: most obviously Joseph Sisko’s restaurant in New Orleans and Chateau Picard, evidencing that not just small possessions are allowed but that the land itself is still privately owned. One could argue that these aren’t really Sisko and Picard’s to own, but they are routinely referred to as “his” restaurant and vineyard so we gotta go with Occam’s Razor here and assume they do, in fact, own them.

It’s important to pay attention to the specificity of these cases. Chateau Picard is essentially a heritage vineyard, deliberately eschewing modern production techniques to deliver the authentic French wine experience. The same is true, in a more down-home way, of Sisko’s restaurant in New Orleans. If you think about the modern economy, highly efficient highly rationalized food service firms (Olive Garden or TGI Friday’s) exist along side organic locally-sourced farm-to-table operations.

The central conceit of Trek is that technology gets better and better, so things that are mass produced and rationalized get cheaper and more abundant. So there’s a post-scarcity economy where anyone can replicate any kind of consumer goods he wants. Webb sees a welfare state, but I actually see something different. It’s simply that energy is abundant enough that people have unrestricted access to consumer-grade replicators. Under the circumstances nobody needs to work to survive and there’s really no point in maintaining a cash economy. But by definition improved technology can’t increase the efficiency of historical production techniques. If the promise of Sisko’s is a home-cooked New Orleans meal, then Sisko’s can’t partake in the post-scarcity economy. Similarly, you can replicate wine in unlimited quantities but a Chateau Picard vintage is by definition a scarce commodity. People appear to operate these businesses for roughly the same reason that Starfleet officers cruise around the galaxy—for a sense of personal fulfillment rather than enrichment. The Federation has clearly acted so as to prevent the existence of any kind of meaningful banking system, and though various mediums of exchange seem to be floating around there isn’t enough stuff for sale for people to really focus on it as an issue.

So what do the producers of scarce goods do? Well, presumably they’re giving a lot of stuff away. Friends and family get bottles of wine. Perhaps you send a case or two to some particularly admired athletes or scientists or other heroes. Maybe artisanal wine just isn’t that popular in general. And maybe you barter some bottles for other artisanal goods. Maybe you have a friend who hand-carves furniture. But at its most fundamental level, it’s a gift economy. The point of running your restaurant or your vineyard is essentially to show off your mastery, not accumulate wealth. There may be some more-or-less formal exchanges, but the key point is to get the output into people’s hands and not work so hard as to make yourself miserable.

We can imagine that Federation Credits exist primarily to let people consume government-provided by scarce resources. Housing, interstellar transportation, child and elder care, energy-intensive capital goods for your hobby/business. This is not a currency per se. It exists to ensure that there isn’t wild overconsumption of goods that are nevertheless intended to be generally available. The Federation probably also uses them to facilitate transactions with other cultures. A non-Federation individual or organization who performs some useful service gets “Credits” entitling him to claim Federation energy or logistical services in the future. Despite official propaganda to the contrary, these credits do circulate as a kind of money in private society. But given the absence of banking, the uselessness of credits for obtaining consumer goods, social stigmatization of wealth accumulation, and the fact that it would generally be considered insulting to offer someone money in exchange for labor (just as today you can pay someone money for sex, but you’d be very careful before making the offer even in places where it’s perfectly legal) it’s not generally circulating in this way.